where \((_{{x}}^t, _{{\lambda }}^t)\)  is a martingale difference with \(\mathbb {E}[(_{{x}}^t, _{{\lambda }}^t) \mid \mathcal {F}_{t-1}] = {0}\) , brought by generating a sample \(\xi _t\sim {\mathcal {P}}\)  and solving Newton systems via randomized solvers; and \((_{{x}}^t, _{{\lambda }}^t)\)  is the remaining error term.
Compared to the studies on unconstrained ASGD and stochastic Newton [1]}, [2]}, [3]}, [4]}, the two main components of AI-StoSQP—adaptivity and inexactness—kick in all terms in the recursion (REF ), and they lead to a much more challenging analysis. In short, we have to deal with a random stepsize \({\bar{\alpha }}_t\) , and \(^t = (_{{x}}^t, _{{\lambda }}^t)\)  and \(^t = (_{{x}}^t, _{{\lambda }}^t)\)  also contain the approximation error from randomized solvers.
